282 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
This solution may be put under the form 
w={P<p( < r)}(^ 1 +a 1 — 1 )(*!+<*!— 3) . . (n+%+2) | — — ^ ~ } » 
the interpretation of 7r 1 being 
= ^(*0 +hf(x—\). 
The method of this section will, I believe, be found to succeed in every known 
integrable case, while it includes some unknown before, one of which is the most 
general yet discovered. 
Postscript. 
The general rule for the integration of linear differential equations in series, B. § 2, 
requires in a particular case to be slightly modified. If y o (D) involves a pair of 
factors such as <p(D), <p(D+r), r being a multiple of the common difference of 
the values of m, we must in the equation of / 0 (D)w=0 write |<p(D)} 2 in place of 
<p(D)<p(D+r), and similarly when there are more such factors. Thus, in the equa- 
tion D(D+2)w— qh 2e u=Q, the form of the assumption for u will be determined by 
the equation D^=0, and not by D(D-f-r)w=0. A slight change of expression would 
make the rule as general as the principle on which it is founded, and reduce to its 
dominion every case in which a linear differential equation can be integrated by 
ascending or descending developments. The theory of series infinite in both direc- 
tions still remains to be examined. 
Fearful of extending this paper beyond its due limits, I have abstained from intro- 
ducing any researches not essential to the development of that general method in 
analysis which it was proposed to exhibit. It may however be remarked that the 
principles on which the method is founded have a much wider range. They may be 
applied to the solution of functional equations, to the theory of expansions, and, to a 
certain extent, to the integration of non-linear differential equations. The position 
which I am most anxious to establish is, that any great advance in the higher ana- 
lysis must be sought for by an increased attention to the laws of the combinations 
of symbols. The value of this principle can scarcely be overrated ; and I only regret 
that in the absence of books, and under circumstances unfavourable for mathematical 
investigation, I have not been able to do that justice to it in this essay which its im- 
portance demands. 
Lincoln, August 31, 1844. 
